The function F(x) =
∑xj≤xf(xj)is called the cumulative frequency function asso-
ciated with the discrete sample. In the continuous case it is called the distribution
function F(x) and calculated by the integral F(x) =
∫x−∞f(x)dx which represents
the area from −∞ to xunder the probability density function. These summation
processes are illustrated in the figure 11-6.
Figure 11-6. Cumulative frequency function for discrete and continuous case.
In both the discrete and continuous cases the cumulative frequency function
represents the probability
F(x) = P(X≤x) =∫x−∞f(x)dx with 1 −F(x) = P(X > x ) =
∫∞xf(x)dx (11 .39)with the property that if a, b are points xwith a < b, then
P(a < x ≤b) = P(X≤b)−P(X≤a) = F(b)−F(a) (11 .40)which represents the probability that a random variable X lies between aand b. In
the discrete case
P(a < X ≤b) =∑a<xj≤bf(xj) = F(b)−F(a) (11 .41)and in the continuous case
P(a < X ≤b) =∫baf(x)dx =F(b)−F(a) (11 .42)